Stability of Waves With Continuous Spectrum

Ohio State University Research Foundation -Do Not Use, Columbus OH

Investigators

Abstract

NSF Award Abstract - DMS-0204072 Mathematical Sciences: Stability of Waves with Continuous Spectrum Abstract 0204072 Oh Periodic waves and spiral waves arise in a rich variety of applied problems. Important examples include periodic wave patterns in models of media with multiple phases, for example van der Waals gas dynamics and elasticity, or three-phase flow in porous media. Planar spiral wave patterns occur in chemical reactions, such as the Belousov-Zhabotinsky reaction, and in various biological and physical systems. There is a similarity between periodic waves and spiral waves, since the continuous spectrum associated with a periodic wave or a spiral wave touches the imaginary axis. The goal of this project is to investigate linear and nonlinear stability of such waves in not only the one-dimensional case but also multi-dimensional cases. Mathematical models of interest in physics, chemistry, biology, and engineering often have classes of special solutions that are particularly important for understanding behavior of the systems under study. This project analyzes two such important classes of solutions in models that involve partial differential equations: periodic waves and spiral waves. The goal is to provide rigorous mathematical analysis of the stability of these solutions. This qualitative information determines the suitability of the differential equations to model the physical phenomena under investigation.